Logarithmic Voronoi Cells for Gaussian Models

TL;DR

This paper extends the concept of logarithmic Voronoi cells to Gaussian models, revealing their structure, relationships, and properties, including explicit descriptions for certain models and conjectures for others.

Contribution

It introduces the theory of logarithmic Voronoi cells for Gaussian models, compares them with log-normal spectrahedra, and provides explicit characterizations for specific covariance models.

Findings

Logarithmic Voronoi cells coincide with log-normal spectrahedra for models of ML degree one.

Explicit semi-algebraic descriptions are provided for bivariate correlation models.

A conjecture is proposed that these cells are not semi-algebraic in unrestricted correlation models.

Abstract

We extend the theory of logarithmic Voronoi cells to Gaussian statistical models. In general, a logarithmic Voronoi cell at a point on a Gaussian model is a convex set contained in its log-normal spectrahedron. We show that for models of ML degree one and linear covariance models the two sets coincide. In particular, they are equal for both directed and undirected graphical models. We introduce decomposition theory of logarithmic Voronoi cells for the latter family. We also study covariance models, for which logarithmic Voronoi cells are, in general, strictly contained in log-normal spectrahedra. We give an explicit description of logarithmic Voronoi cells for the bivariate correlation model and show that they are semi-algebraic sets. Finally, we state a conjecture that logarithmic Voronoi cells for unrestricted correlation models are not semi-algebraic.